Matrix Properties

Matrices are rectangular arrays of numbers that follow a rich set of algebraic rules governing how they can be added, scaled, and multiplied. Understanding these properties is essential because they tell us exactly when we can rearrange calculations, when order matters, and how matrix operations relate to the familiar arithmetic of real numbers. Many properties of matrix algebra mirror ordinary arithmetic: matrix addition is commutative and associative, scalar multiplication distributes over addition, and there exist special matrices like the zero matrix and the identity matrix that play the same roles as 0 and 1 in real-number arithmetic. However, matrix multiplication introduces crucial differences — most importantly, it is generally not commutative, meaning AB and BA can be completely different matrices or one may not even be defined. Mastering these properties allows you to simplify complex expressions, solve systems of equations efficiently, and build the foundation for advanced topics such as determinants, eigenvalues, and linear transformations. This article surveys the key algebraic properties of matrix addition, scalar multiplication, and matrix multiplication, with precise statements and worked examples.

Key takeaways
  • Matrix Addition Properties
  • Scalar Multiplication Properties
  • Matrix Multiplication: When It Is Defined
  • Matrix Multiplication Is NOT Commutative
  • Matrix Multiplication Is Associative and Distributive
  • The Identity Matrix and Transpose Properties

Matrix Addition Properties

For matrices A, B, and C of the same size m×n, addition satisfies four key properties. Commutativity: A + B = B + A. Associativity: (A + B) + C = A + (B + C). Additive identity: there exists a zero matrix 0 (all entries 0) such that A + 0 = A. Additive inverse: for every A there exists −A (negate every entry) such that A + (−A) = 0. These mirror the properties of real-number addition exactly.

Scalar Multiplication Properties

If c and d are real scalars and A, B are m×n matrices, scalar multiplication satisfies: distributivity over matrix addition, c(A + B) = cA + cB; distributivity over scalar addition, (c + d)A = cA + dA; associativity with scalars, c(dA) = (cd)A; and identity, 1·A = A. Scalar multiplication simply multiplies every entry of the matrix by the scalar, so these properties follow directly from real-number arithmetic applied entry-by-entry.

Matrix Multiplication: When It Is Defined

The product AB is defined only when the number of columns of A equals the number of rows of B. If A is m×n and B is n×p, then AB is m×p. The (i,j) entry of AB is computed as the dot product of row i of A with column j of B: (AB)_{ij} = ∑_{k=1}^{n} a_{ik} b_{kj}. This size-matching requirement means that even if AB is defined, BA may not be, and when both are defined they may have different sizes.

Matrix Multiplication Is NOT Commutative

This is the single most important distinction from real-number algebra. In general, AB ≠ BA even when both products are defined and have the same size. For example, with 2×2 matrices it is easy to find cases where AB and BA give completely different results. When AB = BA for a specific pair, we say A and B commute, but this is a special circumstance, not the rule. Always be careful about the order of factors when manipulating matrix expressions.

Matrix Multiplication Is Associative and Distributive

Although not commutative, matrix multiplication IS associative: (AB)C = A(BC) whenever the products are defined. This means you can regroup factors without changing the result, but you cannot swap them. Distributivity also holds: A(B + C) = AB + AC (left distributivity) and (B + C)A = BA + CA (right distributivity). Note that both forms must be listed separately because order matters. Also, c(AB) = (cA)B = A(cB) for any scalar c.

The Identity Matrix and Transpose Properties

The n×n identity matrix I_n has 1s on the main diagonal and 0s elsewhere. For any m×n matrix A: I_m · A = A and A · I_n = A. The transpose A^T switches rows and columns: (A^T)_{ij} = A_{ji}. Key transpose properties are (A^T)^T = A, (A + B)^T = A^T + B^T, (cA)^T = c A^T, and critically (AB)^T = B^T A^T — note the reversal of order.

The Key Equation

(AB)T=BTAT(AB)^T = B^T A^T
  • A — Matrix A (dimensionless): An m×n matrix; after transposing, A^T is n×m
  • B — Matrix B (dimensionless): An n×p matrix; after transposing, B^T is p×n
  • AB — Matrix product (dimensionless): The m×p matrix obtained by multiplying A and B
  • (AB)^T — Transpose of the product (dimensionless): The p×m matrix obtained by transposing the product AB
  • B^T A^T — Reversed product of transposes (dimensionless): The product of B^T (p×n) and A^T (n×m), giving a p×m matrix — same size as (AB)^T

Worked Example

Problem. Let A = [[1, 2], [0, 3]] and B = [[4, 1], [2, 5]]. Verify that AB ≠ BA, then confirm the associative law using C = [[1, 0], [0, 2]].

AB=(811615)≠(411219)=BA,A(BC)=(AB)C=(822630)AB = \begin{pmatrix}8 & 11\\6 & 15\end{pmatrix} \neq \begin{pmatrix}4 & 11\\2 & 19\end{pmatrix} = BA, \quad A(BC) = (AB)C = \begin{pmatrix}8 & 22\\6 & 30\end{pmatrix}

Answer. AB = [[8,11],[6,15]] ≠ [[4,11],[2,19]] = BA (multiplication is NOT commutative), and A(BC) = (AB)C = [[8,22],[6,30]] (multiplication IS associative).

Figure
Diagram
Matrix Multiplication: AB vs BA (Non-Commutativity)A = 2×3, B = 3×4AB is defined:cols(A) = rows(B) = 3 ✓Result: 2×4 matrixBA: cols(B)=4 ≠ rows(A)=2BA is NOT defined ✗A = 2×2, B = 2×2Both AB and BA defined.AB = [[8,11],[6,15]]BA = [[4,11],[2,19]]AB ≠ BA in general!Properties Summary TablePropertyAdditionMultiplicationCommutative✓ A+B = B+A✗ AB ≠ BAAssociative✓✓ (AB)C=A(BC)Distributive✓✓ A(B+C)=AB+AC
Origins · 1 minArthur Cayley and how this was discovered
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Arthur Cayley
1821–1895 · British

Arthur Cayley is credited with developing the modern theory of matrices in his landmark 1858 paper 'A Memoir on the Theory of Matrices.' He was the first to treat matrices as algebraic objects in their own right, define matrix multiplication, identify the identity matrix, and prove results such as the Cayley–Hamilton theorem (which states that every square matrix satisfies its own characteristic equation). His formalization of the properties of matrix operations laid the groundwork for all of linear algebra. Cayley was a practicing lawyer for fourteen years while simultaneously producing hundreds of mathematical papers on the side — he eventually gave up law to become a professor of mathematics at Cambridge, where he flourished as a prolific researcher.

  • Matrix Operations — Matrix properties only make sense in the context of the operations (addition, scalar multiplication, matrix multiplication) they govern; these topics are inseparable.
  • Determinant of a Matrix — The determinant uses matrix properties heavily; for instance, det(AB) = det(A)·det(B) is a key multiplicative property that connects to invertibility.
  • Matrix Inverse — The inverse matrix relies on properties like the identity matrix and associativity: if AB = I then B = A⁻¹, the unique matrix undoing A.
  • Linear Transformations — Every linear transformation can be represented as a matrix, and the algebraic properties of matrices correspond directly to properties of these transformations.
  • Systems of Linear Equations — Matrix properties are used to manipulate the equation Ax = b, enabling systematic solution methods like Gaussian elimination and matrix inversion.